Summary
This paper outlines a Boundary Element Method (BEM) for a piece-wise
analytic solution of the Laplace (Poisson) equation for pseudosteady-state,
single-phase flow on unstructured, rectangular grids. The method models
flow through a reservoir that has been segmented into interacting homogeneous
rectangular regions; no further discretization of the solution space analogous
to grid refinement in numerical schemes is required for improved
accuracy. Rather, boundary discretization allows for continuation of
pressure and flux. Previous work on pressure distribution modeling is
extended to analytically capture the stream function. Stream-function solutions
can then form the basis for other performance measures, such as improved oil
recovery efficiency estimation or tracer flow analysis. Moving beyond
structured grids into unstructured grid geometry allows for advanced
flexibility in problem development and improved efficiency in solution
construction. The analytic approach avoids the need for numerical
differentiation of the pressure field and particle tracking methods to recover
streamlines. Capturing flow in highly heterogeneous media, without local
grid refinement, is demonstrated to showcase the robustness of the technique in
handling complex reservoir architecture, of particular interest in optimal well
positioning and optimal well-pattern development.
Introduction
The solution to fluid flow problems is typically a map of the driving force
(i.e., potential or pressure). A more intuitive result is a map that
shows actual trajectories of fluid elements, the stream function (Muskat
1937). Potential, F, and stream-function, Ψ, are related in 2D by their
spatial gradients.
(Eq. 1)
Curves of constant stream-function value are the so-called
streamlines. Stream-function indexing is associated with integration, as
the difference in stream-function indices represents the amount of fluid
flowing between streamlines of those fixed values. By definition, no
fluid convection occurs across a streamline. Unfortunately, the concept
of stream function is restricted to two dimensions. While an orthogonal
mesh to the pressure field can certainly be constructed in 3D, an equivalent of
stream function is found in three dimensions only for flows exhibiting symmetry
properties that effectively reduce the dimensionality of the flow problem.
© 2007. Society of Petroleum Engineers
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History
- Original manuscript received:
14 July 2005
- Revised manuscript received:
26 January 2007
- Manuscript approved:
1 February 2007
- Version of record:
20 June 2007
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